Hypernumbers and Other Exotic Stuff
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1958: Photopic Field Theory for Natural Hydroso1s
UC San Diego SIO Reference Title Photopic field theory for natural hydroso1s Permalink https://escholarship.org/uc/item/0dk1r645 Author Preisendorfer, Rudolph W Publication Date 1958-09-10 eScholarship.org Powered by the California Digital Library University of California Visibility Laboratory- University of California Scripps Institution of Oceanography San Diego 52, California PHOTIC FIELD THEORY FOR NATURAL HYDROSOLS R. W. Preisendorfer 10 September 1958 Bureau of Ships Index Number NS 714-100 Contract NObs-72092 SIO REFERENCE 58-66 Approved: Approved for Distribution: /£-*.*-'A Seibert Q. Duntley, Director ^—> Roger Revelle, Director Visibility Labora/tory Scripps Institution of Oceanography 1 Photic Field Theory for Natural Hydrosols 2 Rudolph W. Preisendorfer INTRODUCTION The purpose of this note is to present an example of the application of the vector theory of the photic field ("light field") to an important class of scattering-absorbing optical media, namely the class of natural hydrosols consisting, e.g., of oceans, harbors, and lakes. The application is at the same time of practical value in that it yields explicit expressions for the depth-dependence of the light vector in terms of its components at the surface and certain of the optical properties of these media. Furthermore, the discussion presents particularly simple interpretations of the quasipotential and related functions. These interpretations emerge naturally from the geometry and physics of the present application. In this way we add to the evidence that the formalism of the photic field as developed by Moon, Spencer, and others (1), (2), * is of more than academic This paper represents results of research which has been supported by the Bureau of Ships, U. -
An Elementary Approach to Boolean Algebra
Eastern Illinois University The Keep Plan B Papers Student Theses & Publications 6-1-1961 An Elementary Approach to Boolean Algebra Ruth Queary Follow this and additional works at: https://thekeep.eiu.edu/plan_b Recommended Citation Queary, Ruth, "An Elementary Approach to Boolean Algebra" (1961). Plan B Papers. 142. https://thekeep.eiu.edu/plan_b/142 This Dissertation/Thesis is brought to you for free and open access by the Student Theses & Publications at The Keep. It has been accepted for inclusion in Plan B Papers by an authorized administrator of The Keep. For more information, please contact [email protected]. r AN ELEr.:ENTARY APPRCACH TC BCCLF.AN ALGEBRA RUTH QUEAHY L _J AN ELE1~1ENTARY APPRCACH TC BC CLEAN ALGEBRA Submitted to the I<:athematics Department of EASTERN ILLINCIS UNIVERSITY as partial fulfillment for the degree of !•:ASTER CF SCIENCE IN EJUCATION. Date :---"'f~~-----/_,_ffo--..i.-/ _ RUTH QUEARY JUNE 1961 PURPOSE AND PLAN The purpose of this paper is to provide an elementary approach to Boolean algebra. It is designed to give an idea of what is meant by a Boclean algebra and to supply the necessary background material. The only prerequisite for this unit is one year of high school algebra and an open mind so that new concepts will be considered reason able even though they nay conflict with preconceived ideas. A mathematical science when put in final form consists of a set of undefined terms and unproved propositions called postulates, in terrrs of which all other concepts are defined, and from which all other propositions are proved. -
Semilattice Sums of Algebras and Mal'tsev Products of Varieties
Mathematics Publications Mathematics 5-20-2020 Semilattice sums of algebras and Mal’tsev products of varieties Clifford Bergman Iowa State University, [email protected] T. Penza Warsaw University of Technology A. B. Romanowska Warsaw University of Technology Follow this and additional works at: https://lib.dr.iastate.edu/math_pubs Part of the Algebra Commons The complete bibliographic information for this item can be found at https://lib.dr.iastate.edu/ math_pubs/215. For information on how to cite this item, please visit http://lib.dr.iastate.edu/ howtocite.html. This Article is brought to you for free and open access by the Mathematics at Iowa State University Digital Repository. It has been accepted for inclusion in Mathematics Publications by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Semilattice sums of algebras and Mal’tsev products of varieties Abstract The Mal’tsev product of two varieties of similar algebras is always a quasivariety. We consider when this quasivariety is a variety. The main result shows that if V is a strongly irregular variety with no nullary operations, and S is a variety, of the same type as V, equivalent to the variety of semilattices, then the Mal’tsev product V ◦ S is a variety. It consists precisely of semilattice sums of algebras in V. We derive an equational basis for the product from an equational basis for V. However, if V is a regular variety, then the Mal’tsev product may not be a variety. We discuss examples of various applications of the main result, and examine some detailed representations of algebras in V ◦ S. -
YET ANOTHER SINGLE LAW for LATTICES 1. Introduction Given A
YET ANOTHER SINGLE LAW FOR LATTICES WILLIAM MCCUNE, R. PADMANABHAN, AND ROBERT VEROFF Abstract. In this note we show that the equational theory of all lattices is defined by the single absorption law (((y x) x) (((z (x x)) (u x)) v)) (w ((s x) (x t))) = x: _ ^ _ ^ _ _ ^ ^ ^ _ _ ^ _ This identity of length 29 with 8 variables is shorter than previously known such equations defining lattices. 1. Introduction Given a finitely based equational theory of algebras, it is natural to determine the least number of equations needed to define that theory. Researchers have known for a long time that all finitely based group theories are one based [2]. Because of lack of a cancellation law in lattices (i.e., the absence of some kind of a subtraction operation), it was widely believed that the equational theory of lattices cannot be defined by a single identity. This belief was further strengthened by the fact that two closely related varieties, semilattices and distributive lattices, were shown to be not one based [9, 7]. In the late 1960s researchers attempted to formally prove that lattice theory is not one based by trying to show that no set of absorption laws valid in lattices can capture associativity. Given one such attempt [6], Padmanabhan pointed out that Sholander's 2-basis for distributive lattices [10] caused the method to collapse. This failure of the method led to a proof of existence of a single identity for lattices [7]. For a partial history of various single identities defining lattices, their respective lengths, and so forth, see the latest book on lattice theory by G. -
``Mathematics'' and ``Physics'' in the Science of Harmonics
NISSUNA UMANA INVESTIGAZIONE SI PUO DIMANDARE VERA SCIENZIA S’ESSA NON PASSA PER LE MATEMATICHE DIMOSTRAZIONI LEONARDO DA VINCI vol. 4 no. 3-4 2016 Mathematics and Mechanics of Complex Systems STEFANO ISOLA “MATHEMATICS” AND “PHYSICS” IN THE SCIENCE OF HARMONICS msp MATHEMATICS AND MECHANICS OF COMPLEX SYSTEMS Vol. 4, No. 3-4, 2016 dx.doi.org/10.2140/memocs.2016.4.213 ∩ MM “MATHEMATICS” AND “PHYSICS” IN THE SCIENCE OF HARMONICS STEFANO ISOLA Some aspects of the role that the science of harmonics has played in the history of science are discussed in light of Russo’s investigation of the history of the concepts of “mathematics” and “physics”. 1. The rambling route of the ancient scientific method In several places in Russo’s writings on the history of science, one can find en- lightening discussions about the meanings of the concepts of “physics” and “math- ematics”, along with the particular notions of truth involved in them; see, e.g., [58, Chapter 6.6; 60, Chapter 15; 56; 57]. Both terms derive from the Greek: the original meaning of the former was the investigation of everything that lives, grows or, more generally, comes into existence, whereas the latter referred to all that is studied, thus deriving its meaning not from its content but from its method. In the Hellenistic period, the term “physics” continued to be used to indicate that sector of philosophy that addressed nature (the other sectors being ethics and logic), thus corresponding to what came to be called “natural philosophy” in modern times. On the other hand, the term “mathematics” was used to indicate all the disciplines (including geometry, arithmetic, harmonics, astronomy, optics, mechanics, hydro- statics, pneumatics, geodesy and mathematical geography) that shared the same method of investigation, based on the construction of theories by which “theorems” are proved, leaning on explicitly stated initial assumptions. -
New Zealand Number 114 Summer 2014 Skeptic
New Zealand Number 114 Summer 2014 Skeptic Mathematics and Pseudoscience A mathematician’s experiences with mathematical cranks The TPP and its Impact on the NZ Health Sector ACC and Acupuncture Mark Hanna investigates skeptics.nz New Zealand Skeptics CONTENTS ABOUT US 3 Editorial The New Zealand Skeptics form a network of New Zealanders including 4 Newsfront scientists, health professionals, teachers, magicians and many others from all walks of life. Members have 6 Letters a variety of religious faiths, economic beliefs and political leanings, but are all 7 Mathematics and Pseudoscience interested in examining what objective Steven Galbraith writes about his experiences scientific support there is for claims with mathematical cranks of such things as psychic abilities, alternative health practices, creationism and other areas where science, pseudo- 12 The Trans-Pacific Partnership and science and shonky science interact. its Impact on the NZ Health Sector 14 A tribute to Warwick Don CONTRIBUTIONS Contributions are welcome and should be 15 BioBlog by Alison Campbell sent to: P.O. Box 30501 17 Science-Based Healthcare Lower Hutt Mark Hanna investigates the ACC and 5040 acupuncture email: [email protected] Deadline for next issue: 21 Complaining Cheat Sheet 10 April 2015 23 Science-Based Medicine Letters for the Forum may be edited Steven Novella talks about the latest bogeyman as space requires – up to 250 words is preferred. 26 Infectious Thoughts by Siouxsie Wiles Please indicate the publication and date of all clippings for Newsfront. 27 The Humanist Material supplied by email or CD is 28 The Loose Change Range appreciated. Luke Oldfield discusses the art of engaging with a ‘Non-Opinion’ Permission is given to other non-profit skeptical organisations to reprint material from this publication, provided the author 30 Skeptacular! by Mark Maultby and NZ Skeptic Inc. -
Lattices Without Absorption
LATTICES WITHOUT ABSORPTION A. B. ROMANOWSKA Warsaw University of Technology Warsaw, Poland (joint work with John Harding New Mexico State University Las Cruces, New Mexico, USA) 1 BISEMILATTICES A bisemilattice is an algebra (B; ·; +) with two semilattice operations · and +, the first inter- preted as a meet and the second as a join. A Birkhoff system is a bisemilattice satisfying a weakened version of the absorption law for lattices known as Birkhoff's equation: x · (x + y) = x + (x · y): Each bisemilattice induces two partial order- ings on its underlying set: x ≤· y iff x · y = x; x ≤+ y iff x + y = y: 2 EXAMPLES Lattices: x + xy = x(x + y) = x, and ≤·=≤+ . (Stammered) semilattices: x · y = x + y, and ≤·=≥+ . Bichains: both meet and join reducts are chains, e.g. 2-element lattice 2l, 2-element semilattice 2s, and the four non-lattice and non-semilattice 3-element bichains: 3 1 3 2 3 3 3 2 2 3 2 1 2 1 2 3 1 2 1 3 1 2 1 1 3d 3n 3j 3m 3 EXAMPLES, cont. Meet-distributive Birkhoff systems: x(y + z) = xy + xz (MD), e.g. 3m. Join-distributive Birkhoff systems: x + yz = (x + y)(x + z) (JD), e.g. 3j. Distributive Birkhoff systems: satisfy both (MD) and (JD), e.g. 3d. Quasilattices: (x + y)z + yz = (x + y)z (mQ), (xy + z)(y + z) = xy + z (jQ), or equivalently: x + y = x ) (xz) + (yz) = xz, xy = x ) (x + z)(y + z) = x + z. 4 SEMILATTICE SUMS Each Birkhoff system A has a homomorphism onto a semilattice. -
A Geometry Oã-Nutrition1
A Geometry oÃNutrition1 PARRY MOON AND DOMINA EBERLE SPENCER Electrical Engineering Department, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, and Mathematics Department, University of Connecticut, Starrs, Connecticut 06268 ABSTRACT The nutritional value of a given sample of food may be specified by a nutrition holor in a fictitious 3-space. This holor possesses magnitude and character. In most cases, we are interested primarily in character, and this can be represented by a point in a 2-space (the nutrition triangle). The triangle allows visualization of re lations among foods and also the addition of nutritional values for a combination of foods. J. Nutr. 104: 1535-1542, 1974. INDEXING KEY WORDS geometry •mathematical model •nutrient patterns Downloaded from The worldwide importance of modern acter of a food may be specified by giving nutritional theory can hardly be overesti any two of the above coordinates. These mated. Nutrition calculations, however, coordinates can be plotted in a nutrition have remained in the arithmetic stage and triangle (fig. 1). have not utilized the possibilities of alge The diagram is advantageous because it braic and geometric formulation. In par shows at a glance the relations among jn.nutrition.org ticular, a geometrization of nutrition al foods: which food has the higher lipid lows one to visualize the relations among content, which has the more protein, etc. nutrients and thus to bypass much of the The triangle can be divided into regions. present routine computation, in accordance Sugars, syrups, and honey are almost pure with the principle that a picture is worth carbohydrate, so they are represented by by on November 25, 2009 a thousand words. -
Laws of Boolean Algebra and Boolean Algebra Rules
4/10/2020 Laws of Boolean Algebra and Boolean Algebra Rules Home / Boolean Algebra / Laws of Boolean Algebra Laws of Boolean Algebra Boolean Algebra uses a set of Laws and Rules to define the operation of a digital logic circuit As well as the logic symbols “0” and “1” being used to represent a digital input or output, we can also use them as constants for a permanently “Open” or “Closed” circuit or contact respectively. A set of rules or Laws of Boolean Algebra expressions have been invented to help reduce the number of logic gates needed to perform a particular logic operation resulting in a list of functions or theorems known commonly as the Laws of Boolean Algebra. Boolean Algebra is the mathematics we use to analyse digital gates and circuits. We can use these “Laws of Boolean” to both reduce and simplify a complex Boolean expression in an attempt to reduce the number of logic gates required. Boolean Algebra is therefore a system of mathematics based on logic that has its own set of rules or laws which are used to define and reduce Boolean expressions. The variables used in Boolean Algebra only have one of two possible values, a logic “0” and a logic “1” but an expression can have an infinite number of variables all labelled individually to represent inputs to the expression, For example, variables A, B, C etc, giving us a logical expression of A + B = C, but each variable can ONLY be a 0 or a 1. Examples of these individual laws of Boolean, rules and theorems for Boolean Algebra are given in the following table. -
2.2 Set Operations 127
P1: 1 CH02-7T Rosen-2311T MHIA017-Rosen-v5.cls May 13, 2011 10:24 2.2 Set Operations 127 2.2 Set Operations Introduction Two, or more, sets can be combined in many different ways. For instance, starting with the set of mathematics majors at your school and the set of computer science majors at your school, we can form the set of students who are mathematics majors or computer science majors, the set of students who are joint majors in mathematics and computer science, the set of all students not majoring in mathematics, and so on. DEFINITION 1 Let A and B be sets. The union of the sets A and B, denoted by A ∪ B, is the set that contains those elements that are either in A or in B, or in both. An element x belongs to the union of the sets A and B if and only if x belongs to A or x belongs to B. This tells us that A ∪ B ={x | x ∈ A ∨ x ∈ B}. The Venn diagram shown in Figure 1 represents the union of two sets A and B. The area that represents A ∪ B is the shaded area within either the circle representing A or the circle representing B. We will give some examples of the union of sets. EXAMPLE 1 The union of the sets {1, 3, 5} and {1, 2, 3} is the set {1, 2, 3, 5}; that is, {1, 3, 5}∪{1, 2, 3}={1, 2, 3, 5}. ▲ EXAMPLE 2 The union of the set of all computer science majors at your school and the set of all mathe- matics majors at your school is the set of students at your school who are majoring either in mathematics or in computer science (or in both). -
Views of Farms, Residences, Mills &C., Portraits of Well-Known Citizens, and the Official County Map
Donald Heald Rare Books A Selection of Rare Books Donald Heald Rare Books A Selection of Rare Books Donald Heald Rare Books 124 East 74 Street New York, New York 10021 T: 212 · 744 · 3505 F: 212 · 628 · 7847 [email protected] www.donaldheald.com California 2017 Americana: Items 1 - 34 Travel and Voyages: Items 35 - 58 Natural History: Items 59 - 80 Miscellany: Items 81 - 100 All purchases are subject to availability. All items are guaranteed as described. Any purchase may be returned for a full refund within ten working days as long as it is returned in the same condition and is packed and shipped correctly. The appropriate sales tax will be added for New York State residents. Payment via U.S. check drawn on a U.S. bank made payable to Donald A. Heald, wire transfer, bank draft, Paypal or by Visa, Mastercard, American Express or Discover cards. AMERICANA 1 ADAMS, Ansel Easton (1902-1984) and Mary Hunter AUSTIN (1868-1934). Taos Pueblo. San Francisco: Grabhorn Press, 1930. Folio (17 x 12 1/2 inches). [6] preliminary pages followed by [14]pp. of text. 12 original mounted photographs, printed on Dessonville paper by Ansel Adams, various sizes to 9 x 6 1/2 inches, each with a corresponding caption leaf. Publisher’s tan morocco backed orange cloth, spine with raised bands in six compartments, marbled endpapers (minor fading to the leather). From an edition of 108 numbered copies signed by the author and the photographer, containing magnificent photographs by Ansel Adams. Possibly the most famous of modern photographic works on the West, Taos Pueblo was a collaboration between the young photographer, Ansel Adams, and one of the most evocative writers on the Southwest, Mary Austin. -
Dissertationes Mathematicae (Rozprawy Matematyczne)
POLSKA AKADEMIA NAUK, INSTYTUT MATEMATYCZNY DISSERTATIONES MATHEMATICAE (ROZPRAWY MATEMATYCZNE) KOMITETREDAKCYJNY BOGDAN BOJARSKI redaktor _ WIESL AW ZELAZKO zaste,pca redaktora ANDRZEJBIAL YNICKI-BIRULA, ZBIGNIEW CIESIELSKI, JERZYL OS,´ ZBIGNIEW SEMADENI CCCIX ARTHUR KNOEBEL and ANNA ROMANOWSKA Distributive multisemilattices W A R S Z A W A 1991 Published by the Institute of Mathematics, Polish Academy of Sciences Typeset in TEX at the Institute Printed and bound by M. & K. Herman, Spokojna 1, Raszyn PRINTEDINPOLAND c Copyright by Instytut Matematyczny PAN, Warszawa 1991 ISBN 83-85116-09-5 ISSN 0012-3862 CONTENTS 1. Introduction . 5 2. Definition, basic examples and properties of multisemilattices . 6 3. The subdirectly irreducibles . 13 4. The lattice of subvarieties of Dn .....................18 5. Subvarieties of Dn defined by identities involving at most two operation symbols . 24 6. Some further comments and open problems . 34 References . 40 Abstract A distributive multisemilattice of type n is an algebra with a family of n binary semilattice operations on a common carrier that are mutually distributive. This concept for n = 2 comprises the distributive bisemilattices (or quasilattices), of which distribu- tive lattices and semilattices with duplicated operations are the best known examples. Multisemilattices need not satisfy the absorption law, which holds in all lattices. Kalman has exhibited a subdirectly irreducible distributive bisemilattice which is neither a lattice nor a semilattice. It has three elements. In this paper it is shown that all the subdirectly irreducible distributive multisemilattices are derived from those for n = 2 simply by duplicating their operations in all possible ways. Thus, up to isomorphism there are 2n −1 of type n, but up to the coarser relation of polynomial equivalence there are only three.